Mathematicians and scientists often express absolute awe — they gawk even — at the power and effectiveness of describing the universe through mathematics. Physicist Eugene Wigner wrote about this in his 1960 piece The Unreasonable Effectiveness of Mathematics in the Natural Sciences saying
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.
Physicists cite Paul Dirac’s 1928 famous equation in quantum mechanics as providing a gift. The equation required the existence of unheard-of “anti-electrons,” that is, electron-like particles with positive energy. And, lo-and-behold, a few years after Dirac published the equation Carl Anderson at Caltech discovered positrons in the physical world. Just recently the mathematics twitter-verse was abuzz with the surprise real-world application of an abstruse piece of mathematics. A century-old result about polynomials proves to be absolutely key to safety protocols for self-driving cars.
So what is it that makes this “unreasonableness” of mathematics so shocking?
This has been a discussion and debate underway for decades, better suited for the scientific philosophers among us. But I do want to address in my naïve way one aspect of this question.
Is the “unreasonableness” of mathematics shocking? Is it shocking to those whose primary encounter of mathematics comes from what is taught in school?
The story of Number
The K-12 curriculum has students explore numbers, their practical arithmetic through to their general structure via algebra. One could, from an advanced perspective, describe that story as follows.
- We, humankind, first learned how to count and so discovered/invented the counting numbers 1, 2, 3, … (maybe add 0 to this list too). These counting numbers come with two natural binary operations, which we call addition and multiplication, and these operations feel tangible and meaningful. This system of counting numbers is practical and is great for solving a number of problems, which in algebraic language translate to finding solutions to equations of the form x + 5 = 7. But, alas, they are not good for solving all such equations. Consider solving x + 20 = 15, for instance.
- So we extend our system of numbers to include solutions to these problematic equations too, that is, we incorporate the additive inverses of counting numbers to obtain the system of integers. What is lovely here that all the properties of addition and multiplication that hold for the counting numbers can be mathematically made to hold in a consistent manner in this setting too — though it is less clear what the practical meaning of these operations are in each case. (Multiplying together two negative numbers, for instance, is unsettling.)
- Nonetheless, this system has still seems to hold meaning for us in many everyday applications. We now have a solution to x + 20 = 15, for instance, which corresponds to the question: The temperature is now 15 degrees. It has warmed up 20 degrees since this morning. What temperature was it then?
- But we’re not out of the woods. There are multiplicative equations too. We can solve 2x = 6 in our system of integers, but we cannot solve 2x = 5, for example. So we extend our system of numbers to include solutions to these types of equations too and create the rationals. And these numbers still seem to have application to the real world in matters of portions and fractions. But what is lovely here too is that all the properties of addition and multiplication that feel good and right to us in the world of counting numbers can be mathematically made to hold in a consistent manner in the world of rational numbers too. (Practical interpretations become murkier still: We can add half a pie to a third of a pie, but what does it mean to multiply two portions of pie?)
- But we’re still not out of the equation-solving woods. Some multiplicative equations such as x² = 4 are solvable in the system of rational numbers, but others, such as x² = 3, are not. So we extend our system of numbers to include solutions to these types of equations as well. We create the reals. And it is possible to extend the notion of addition and multiplication from the rationals to the reals so that all the familiar properties of these operations continue to hold in this context as well. (Though their meaning is much more abstract now: How do you actually multiply two infinitely long decimals? How do you even write one down?)
- But we’re still not out of the woods. Some multiplicative equations still have no solutions: x² =-1, for example. This then leads us to construct the system of complex numbers and mathematically extending the operations of addition and multiplication to this expanded system too. The practical meaning of everything is far less clear, but this is not the worry of the mathematical story: we’re looking for a logical construct of “number” that extends notions of “addition” and “multiplication” in a consistent manner to lead to solutions of equations.
- One might worry that we’re on a never-ending quest here, that with each expansion of our notion of number, we’re going to find new equations that cannot be solved within the system and so require expansion to a new system. But here’s the lovely surprise. In 1806 Jean-Robert Argand proved that, in the case of polynomial equations at least, the story does stop here: Each and every polynomial equation written within the system of complex numbers has a solution within the system of complex numbers. This is called the Fundamental Theorem of Algebra.
Look at what mathematics is doing here. It is taking a practical scenario, the world of counting numbers with two binary operations which have concrete, practical meaning in that system, and letting go of the practical constraints to identify the key, abstract features of that system. Mathematics’ goal is to develop a logically robust system for solving a whole class of equations. It’s theoretical. It’s abstract. It’s based on practicality, but then very gets far removed from it. (And it is beautiful, by the way, too!)
The question of what it means to multiply two negative numbers, two portions of pie, or two infinite decimals is moot. That’s not the story.
The impossible K-12 dilemma
Mathematics, as the fourth letter of STEM, is often seen by the world at large as the auxiliary player in 21-st century education thinking: its purpose is to provide the computational might for the practicalities of science, technology, and engineering. As such, mathematics is expected by the world at large to have practical, tangible answers and so each concept or idea in K-12 mathematics is expected to actually be something real and concrete. Although mathematics might be motivated and inspired by matters practical, its goal is to often attend to the theoretical and philosophical — to identify underlying themes and structures that can be expanded upon and evolved as concepts in their own beautiful right. This mismatch of goals and expectations, I think, comes to particular head in the “What is multiplication?” debates of the past few years.
We first teach multiplication to students in the context of counting numbers where it comes to us as repeated addition. Here 3 x 4 is to be interpreted (in an unsymmetrical manner) as three groups of four objects. If we arrange these groups in rows and look at columns, we then see 4 x 3: four groups of three objects. Interpreting the same picture two different ways shows that our unsymmetrical definition of multiplication here in the world of counting numbers is symmetrical after all. Whoa!
In fact, playing with dots on a page or pebbles on the floor leads us to discover a host of “rules” about multiplication (and about addition) that feel natural and right — and they probably feel so “natural and right” because the counting numbers might be the only set of numbers we humans have a physical feel for. The rules of multiplication include:
a x 1= a (a groups of 1 object is a objects)
a x 0 = 0 (a groups of no objects is no objects)
a x b= b x a (a groups of b objects is the same count as b groups of a objects.)
a x (b+c) = a x b+ a x c (a groups of b+c objects might as well be a groups of b objects and a groups of c objects)
(a x b) x c = a x(b x c) (An a-by-b array of groups, each group with c objects, can be looked at two different ways.)
But mathematicians have extended the idea of number and arithmetic on number to new systems, ones that start to feel less natural. They have shown that each system still possesses two binary operations, + and x, that behave the same way as addition and multiplication do on the counting numbers by following the same list of rules. No real claim is made as to what these binary operations actually are, just how they behave.
In fact, when asked “What is multiplication?” a mathematician might well answer:
Multiplication is any binary operation on a set of objects (numbers) that satisfies the list of rules described above.
It feels like side-stepping the question, but it is not: it is just attending to a different premise of the question. Mathematicians tend to focus on the theoretical structure of arithmetic and recognize that there are many arithmetic systems that have two binary operations that interact according to the same set of rules. Multiplication is — and I mean “is” here — the second of those two binary operations. This is the practical answer in the context of a theoretical question!
But now comes the unreasonableness of mathematics: there are many examples of binary operations in K-12 mathematics that satisfy the same list of basic rules. Here are four.
1.Repeated addition in the world of counting numbers.
2. The area of rectangles in the world of geometry.
3. Expanding and contracting lengths with a scale factor in geometry.
4. Unit conversion.
In the first instance, 3 x 4 is the number 12: the count of objects you have in total when you have three groups of four objects. In the second context, 3 x 4 is the area of a three-by-four rectangle. The answer is not just a number, but a number with units: a 3 feet by 4 feet rectangle has area 3 x 4 = 12 square feet. (Are we allowed to mix units? What is the area of a 3 meter by 4 feet rectangle? Is it 12 meter-feet?) In the third context, 3 x 4 is the result of scaling a quantity by a factor of three. For example, a length of 4 feet becomes a length of 12 feet if we stretch it three times as long. And in the fourth context, using the fact that there are three feet in a yard, we deduce that there are 3 x 4 = 12 feet in four yards.
So in different contexts the answer to 3 x 4 is slightly different: it’s 12 as a count of objects, it is 12 square feet, and it is 12 feet. (And there are also contexts where the units of the answer change completely. For example, 3 tenths times 4 tenths is 12 hundredths.)
There is also a fifth example I should mention.
5. The word “of” in the arithmetic of fractions.
Students are taught to read, for instance, ½ x 6 as “half of six,” that is, to see multiplication as the action of taking a portion of something.
A student will encounter many manifestations of multiplication throughout the K-12 curriculum. But that is what they are: manifestations. They each come from some context where there is a meaningful binary operation that beautifully satisfies the same set of basic rules: the ones listed above. (The unreasonableness of mathematics at play.)
And indeed we can often find links between the different manifestations: subdivide a three-by-four rectangle into twelve unit squares we see an array of squares like we saw an array of dots; we can view repeated the addition problem 3 x 4 as scaling the count of four objects by a factor of three; and so on.
But if we insist on giving the impression that multiplication is one common thing, a single actual concrete thing, and that the links we see are “proof” of that claim, then we might tie our students — and ourselves — into painful intellectual knots.
Well, many actually do say that multiplication is one thing: it is scaling. That is fine. The tricky part then is the changing context of what is being scaled in each scenario: scale the count of apples, scale the area of the picture of an apple, scale the volume of an apple, scale something else. But I am not personally convinced that it is pedagogically appropriate and helpful to insist on this one view. Why not have students identify structure in what they study and observe the repeated structure in these different scenarios? (Sounds like a practice standard of sorts to me.) Isn’t it lovely that one can see that the geometry of rectangles and the use of the word “of” and that stretching and scaling all have the same mathematical structure at play? Why not teach the art of thinking like a mathematician AND the art of applying mathematical structure in multiple contexts, contexts of science, technology, and engineering. Multiplication might be a computation without units, or a computation with units or units squared, or something else.
And what do parents want?
Well, with the “Just teach my kid the math” attitude perhaps, we might argue that parents are calling for the pure mathematicians’ approach! Numbers — the integers, fractions, and decimals — come with a binary operation called multiplication, so … just teach my kid how to multiply! Sounds like an (unwitting) call for sole abstraction to me.